Geometrical construction for Snell's law?

Snell's law from geometrical optics states that the ratio of the angles of incidence $\theta_1$ and of the angle of refraction $\theta_2$ as shown in figure1, is the same as the opposite ratio of the indices of refraction $n_1$ and $n_2$.

$$ \frac{\sin\theta_1}{\sin \theta_2} = \frac{n_2}{n_1} $$

(figure originally from wikimedia)

Now let $P$ be a point in one medium (with refraction index $n_1$) and $Q$ a point in the other one as in the figure. My question is, is there is a nice geometrical construction (at best using only ruler and compass) to find the point $O$ in the figure such that Snell's law is satisfied. (Suppose you know the interface and $n_2/n_1$)?

Edit A long time ago user17762 announced to post a construction. However until now no simple construction was given by anybody. So, does anybody know how to do this?

If you know the interface, then drop perpendiculars from $P$ and $Q$ to the interface. Let the points of intersection be $P'$ and $Q'$. Let $PP' = y_P$ and $QQ' = y_Q$.

Now consider the line segment $P'Q'=x$. You need to find a point $O$ inside $P'Q'$ such that $OP' + OQ' = x$.

Let $OP' = x_P$ and $OQ' = x_Q$.

We now have two equations to solve for $\theta_1$ and $\theta_2$.

$x_P + x_Q = x$ i.e. $$y_P \tan(\theta_1) + y_Q \tan(\theta_2) = x$$

and

$$\frac{\sin(\theta_1)}{\sin(\theta_2)} = \frac{n_2}{n_1}$$.

So the problem is well-defined and hence solving for $\theta_1$ and $\theta_2$ gives $x_1$ and $x_2$.

I shall post the geometric construction later.

To expand on Han de Bruijn's comment: Assume $P=(x,y_1)$ and $Q=(x',-y_2)$ with $x'-x=:d>0$. Then we have to solve the system $$\eqalign{x_1+x_2&=d\cr {n_1x_1\over\sqrt{x_1^2+y_1^2}}&={n_2x_2\over\sqrt{x_2^2+y_2^2}}\cr}$$ for $x_1$ and $x_2$. Introducing $x_2:=d-x_1$ into the squared second equation leads to an equation of degree $4$ for $x_1$ having no obvious solutions in terms of second degree equations. From this we may conclude that there is no ruler and compass construction of the path in question, given the ratio ${n_1\over n_2}$.